Definition df-ushgr

Description: Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function e is an injective (one-to-one) function into subsets of the set of vertices v , representing the (one or more) vertices incident to the edge. This definition corresponds to the definition of hypergraphs in section I.1 of Bollobas p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of Diestel p. 27, where "E is a subset of [... the power set of V, that is the set of all subsets of V" resp. "the elements of E are nonempty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020) (Revised by AV, 8-Oct-2020)

	Ref	Expression
Assertion	df-ushgr	⊢ USHGraph = { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cushgr	⊢ USHGraph
1		vg	⊢ 𝑔
2		cvtx	⊢ Vtx
3	1	cv	⊢ 𝑔
4	3 2	cfv	⊢ ( Vtx ‘ 𝑔 )
5		vv	⊢ 𝑣
6		ciedg	⊢ iEdg
7	3 6	cfv	⊢ ( iEdg ‘ 𝑔 )
8		ve	⊢ 𝑒
9	8	cv	⊢ 𝑒
10	9	cdm	⊢ dom 𝑒
11	5	cv	⊢ 𝑣
12	11	cpw	⊢ 𝒫 𝑣
13		c0	⊢ ∅
14	13	csn	⊢ { ∅ }
15	12 14	cdif	⊢ ( 𝒫 𝑣 ∖ { ∅ } )
16	10 15 9	wf1	⊢ 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } )
17	16 8 7	wsbc	⊢ [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } )
18	17 5 4	wsbc	⊢ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } )
19	18 1	cab	⊢ { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } ) }
20	0 19	wceq	⊢ USHGraph = { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } ) }